Generalizations of string theory

(talk at Generalized Geometry & T-dualities, 5/11/16)

Outline

(feel free to add ``super-" anywhere; always in superspace)

T-theory

  1. Current algebra: strings → double field theories
  2. New spaces: field strengths as gauge fields
  3. Chiral strings: boundary condition → finite spectrum
    1. Effective actions: new for (massive) gravity
    2. Particle amplitudes: scattering eqs. & KLT

(not your father's) F-theory (nor mother's M-theory)

  1. Current algebra: branes → exceptional gravity
  2. 0th-quantized ghosts

T-theory

Current algebra: strings → double field theories 👯

T-duality (worldsheet, no background) (WS '83)
Fields d=1 dimensionally reduced (Buscher '87)
d>1: O(d,d)/O(d)² scalars (Duff '89, Tseytlin '90)
"Double field theory", bosonic & heterotic (WS '93): Our recent work (see MH talk for some details)

Spin Sab & dual Σab (WS '11, Poláček & WS '13)

Type II & RR (Hatsuda, Kamimura, WS '14-'15) 3D Type II off-shell background (Poláček & WS '14)

New spaces: field strengths as gauge fields (WS '11)

T-theory methods generalize to particle field theory, e.g., gravity But also Yang-Mills

Chiral strings: b.c. ⇒ finite spectrum (see BZ talk)

Effective actions (Hohm, Zwiebach, WS '13) Particle amplitudes for all dimensions "Scattering equations" (Cachazo, He, Yuan '13)

From string-like theories (Mason & Skinner '13)

Derivation from usual string theories (WS '15)

KLT factorization (Huang, Yuan, WS '16)

F-theory (& M-theory) (Linch & WS '15)

Before branes: more-than-DFT for exceptional SG
(Coimbra, Strickland-Constable, Waldram '11;
Berman, Cederwall, Kleinschmidt, Thompson '12)

Current algebra on fundamental branes

0th-quantized ghosts (WS '16)

Appendix: Some technical details

T-theory current algebra with Ramond-Ramond

Currents: (Sab,Dα,Paαab)±αα'αα'

{D±,D±} = P±δ, {D+,D} = Υδ

[D±,P±] = Ω±δ, [D±,Ϝ] = Ωδ

[S±,not Σ] ~ not Σ, [S±±] = Σ±δ + δ'

[P±,P±] = Σ±δ + δ', [Υ,Ϝ] = (Σ+)δ + δ'

{D±±} = Σ±δ + δ'

1st-order Yang-Mills as Chern-Simons

With torsion [dA,dB} = TABCdC, the CS form is

XABC = ½A[AdBAC) −¼A[ATBC)DAD +⅓iA[AABAC)

In particular, for

[pa,pb] = σab ⇒ Ta,bcd = ½δ[acδb]d

we choose

L = ½aa,b,cdXa,b,cd , aa,b,cd = ηa[cηd]b

Then, labeling Ap→A, Aσ→F,

L ~ AσA + FpA − FF + FAA

gives the usual 1st-order action after σ→0.

Scattering equation δ from almost-chiral gauge

Change in boundary condition,

Δ = − ln z − ln z̄ → − ln z + ln z̄

followed by change in gauge (with β→∞),

→ − ln z + ln (z̄ − βz) → −
1
β
z

modifies z̄ half of Koba-Nielsen integral

∫dz̄i exp
1
∑ki∙kj
ij
zij
i δ(∑j
ki∙kj
zij
)

KLT for chiral string (toy example, m=0)

Normal → chiral [sin(πt) = −π/Γ(−t)Γ(1+t)]:

Γ(−s)Γ(−t)
Γ(u)
1
Γ(−t)Γ(1+t)
Γ(1−t)Γ(−u)
Γ(1+s)

=
1
s
Γ(−s)Γ(−t)Γ(−u)
Γ(s)Γ(t)Γ(u)

→ −
Γ(−s)Γ(−t)
Γ(u)
1
Γ(−t)Γ(1+t)
Γ(1+t)Γ(+u)
Γ(1-s)

=
1
s

F-theory dimensional reduction & sectioning

σ-VirasoroS ≡ PP
dim. reductionS̊ ≡ pPU ≡ ∂P
sectioningS̥ ≡ ppU̥ ≡ ∂pV ≡ ∂∂

P = bosonic current (selfdual field strength);
p = 0-mode, ∂ for worldvolume

Downward: P→p (otherwise same)

Dimensional reduction kills X's (U = Gauss)

Section conditions kill 0-modes (background) & σ's

(S̊,S̥) = 0 ⇒ M-theory; (U,U̥,V) = 0 ⇒ T-theory

A little F-theory current algebra

d worldvolume coordinates: τ,σM

Spacetime coordinates:Θµ,Xm,Ym',ỸMm'
(32 Θ's, <10 worldvolume scalars Y, their duals Ỹ)

Currents: (S,)Dµ,Pmm'Mm'(,Σ)

[▷M , ▷N} = fMNPPδ + 2iηMNRRδ

⇒ f[MN|QfQ|P)R = 0 = fM(N|QηQ|P]R

SR = ηMNRNM

f's ~ super YM, but doubled spinor indices

Types of F-theories (N=1 D-dim. SG bosons in G/H)

DdHGXσ
13GL(1)GL(2)2+12
24GL(2)SL(3)SL(2)(3,2)(3',1)
36Sp(4)SL(5)105'
411Sp(4,C)SO(5,5)1610
528USp(4,4)E6(6)2727'
6134SU*(8)E7(7)56133

G = "ED+1",
H = covering of SO(D−1,1) with double argument,
H' = SO(10−D)

Y = vector of H', Θ = spinor of H⊗H' (32)